Lecture 04

Lecture 3: Review

• We covered:

  • data wrangling and types of variable names

  • meta data

  • project design

  • summary statistics

  • graphing the mean and standard error graphs

  • pipes and %>% or |> and how to group_by

Our last graph

Lecture 4: How to deal with data wrangling

Introduction to probability distributions

• What is a frequency distribution?
• What is a probability distribution?
• Distributions for variables and for statistics

Estimation

• Populations and samples
• Parameters and statistics

we are going to use some sculpin data that is real!

Wikipedia

Lecture 4: Frequency distributions

Example data we will use will be a combination of data from Toolik Alaska LTER

source

We will specifically look at fishes like

Lake Trout

source

Grayling

Slimy Sculpin

source

Lecture 4: Frequency distributions

• Data - has been cleaned in terms of lake names and species names
• Slimy Sculpin - Toolik Lake

sculpin_df %>% 
  filter(lake == "Toolik") %>% 
  summarize(
    mean = mean(total_length_mm, na.rm = TRUE),
    sd = sd(total_length_mm, na.rm = TRUE),
    se = sd(total_length_mm, na.rm = TRUE)/(sum(!is.na(total_length_mm))^0.5),
    count = sum(!is.na(total_length_mm)), 
    .groups = "drop") 

# A tibble: 1 × 4
   mean    sd    se count
  <dbl> <dbl> <dbl> <int>
1  51.7  12.0 0.834   208

Note in the quarto code we use things to control what we see like
#What does this mean

• # | echo: false
• # | message: false
• # | warning: false
• # | fig-height: 4 # | fig-width: 3
• # | paged-print: false)

Practice Exercise 1: Reading Slimy Sculpin - Toolik Lake

• Data - has been cleaned in terms of lake names and species names
• Slimy Sculpin - Toolik Lake

# Write your code here to read in data
# Remember to use tidy coding skills and comment the HOOI
# 
# 
# library(tidyverse)
# library(patchwork)
# sculpin_df <- read_csv("data/sculpin.csv")
# now look at what is there

Practice Exercise 2: Now lets look at descriptive statistics

Let’s try looking at what the summary of the data tell us

# now do the summary statistics please

Lecture 4: Frequency Distributions

What is a frequency distribution?

• Display of number of observations in certain intervals
• e.g., the number of sculpin per interval in Toolik Lake
• as a table like below or histogram

sculpin_df %>% 
  filter(lake == "Toolik") %>%
  filter(!is.na(total_length_mm)) %>% 
  mutate(length_bin = cut_interval(total_length_mm, length = 2)) %>%
  count(length_bin)

# A tibble: 29 × 2
   length_bin     n
   <fct>      <int>
 1 [10,12]        1
 2 (12,14]        3
 3 (18,20]        1
 4 (22,24]        1
 5 (26,28]        1
 6 (28,30]        1
 7 (30,32]        2
 8 (32,34]        3
 9 (34,36]        4
10 (36,38]        3
# ℹ 19 more rows

Practice Exercise 3: Now try to modify this so it is in 5 mm lenghts

Let’s try looking at what the summary of the data tell us

# now try different bins
sculpin_df %>% 
  filter(lake == "Toolik") %>%
  filter(!is.na(total_length_mm)) %>% 
  mutate(length_bin = cut_interval(total_length_mm, length = 2)) %>%
  count(length_bin)

# A tibble: 29 × 2
   length_bin     n
   <fct>      <int>
 1 [10,12]        1
 2 (12,14]        3
 3 (18,20]        1
 4 (22,24]        1
 5 (26,28]        1
 6 (28,30]        1
 7 (30,32]        2
 8 (32,34]        3
 9 (34,36]        4
10 (36,38]        3
# ℹ 19 more rows

Lecture 4: Frequency Distributions

The alternative is to use a histogram

• the y axis is the count
• the x axis is the bin range
• each bin 0 - 5 and 5 - 10 and 10 - 15 or as you choose
• in ggplot the code looks like

dataframe %>% ggplot(aes(thing_to_count))+
  geom_histogram( 
    binwidth = increments_to_work_with
    )

Practice Exercise 4: This is something you shoudl do

Let’s try stuffing frogs in our pockets

# Write your code here to create funny plot
# Remember to use tidy coding skills and comment the HOOI

Lecture 4: Frequency Distributions

What happens as sample size changes…

• Sampls size
  • Low sample number - 15
  • High sample number - 70
• Frequency distribution takes on “bell-shape”…

Lecture 4: Probability distributions

Can we make assumption about distribution of random variable weight in population?

Probability distribution:

• theoretical frequency distribution in population

Lecture 4: Probability distributions

• For continuous random var: probability density function (PDF)
• PDF: mathematical expression of probabilities associated with getting certain values of random variable
• Area under curve = 1
• i.e., probability of length between 10 and 80 = 1

Lecture 4: Probability distributions

Now we could look at a lot of different ranges of lengths

• probability of the length larger than the mean

• probability of the length larger than 70 mm

• probability of the length between two numbers

Lecture 4: Probability distributions

• Usually need to know probability distribution of random variables in statistical analyses
• Can define many distributions; some do reasonable job especially whit continuous variables
• Different distributions for continuous, discrete variables like a single die

Lecture 4: Probability distributions

Normal (Gaussian): symmetrical, bell-shaped

• Defined in terms of mean and variance (μ, 𝜎2)

• SND (z-distribution) has mean μ=0 , 𝜎2 =1

\(f(y) = \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{(y - \mu)^2}{2\sigma^2}}\)

Lecture 4: Probability distributions

• Lognormal: right-skewed distribution
• Logarithm of random variable is normally distributed
• Common in biology.
• Why would this occur or be common in biology?

Lecture 4: Probability distributions

Binomial (multinomial):

• probability of event that have two outcomes (heads/ tails, dead/alive)
• Defined in terms of “successes” out of set number of trials

In large number of trials: approximately normal distribution

Lecture 4: Probability distributions

Poisson: occurrences of (rare) event in time/space

• E.g., number of
  • Taraxacum officinale - common dandelion in quadrat
  • copepod eaten per minute
  • cells in field of view
• Measures Probability(y= certain integer value)

  • defined in terms of μ or mean

  • Right-skewed at small μ

  • more symmetrical at higher μ

Lecture 4: Data gathering - managing

Also have distributions of test statistics

Test statistics:

• summary values calculated from data used to test hypotheses
• is your result due to chance?

Different test statistics:

• different, well-defined distributions
• allows estimation of probabilities associated with results
• Examples:
  • z-distribution, student’s t-distribution, χ2-distribution, F-distribution

Lecture 4: Samples and populations

Inferential statistics:

• inference from samples to populations

Statistical population:

• All possible observations of interest
• Normally: populations too large to census

Populations are defined in time + space

Examples of statistical populations from you research area?

Lecture 4: Samples and populations

Key characteristic of sample is

• size (n observations; n = sample size)

Characteristics of population - called parameters

• Parameters - Greek letters

Characteristics of samples - statistical estimates of parameters

• statistics= Latin letters

Random sampling crucial for

sample -> population

inference statistics -> parameters

Lecture 4: Parameters and statistics

Two main kinds of summary statistics: - center and spread

Center: - Mean (µ, ȳ): sum of sampled values divided by n - Mode: the most common number in dataset - Median: middle measurement of data; = mean for normal distributions

Mean

\(\mu = \frac{\sum\limits_{i=1}^{n} Y_i}{n}\)

Formula for n odd

\(\text{median = } Y\_{(n+1)/2}\)

Formula for n even

\(\text{median = }\frac{Y_{n/2} + Y_{(n/2)+1}}{ 2}\)

Lecture 4: Parameters and statistics

Spread

• Range: from highest and lowest observation
• Variance (σ2, s2): sum of squared differences of observations from mean, divided by n-1

E.g., fish lengths = 20, 30, 35, 24, 36 g

# A tibble: 1 × 1
   mean
  <dbl>
1    29

\(s^2 = \sum_{i=1}^{n} \frac{(y_i - \bar{y})^2}{n-1}\)

Lecture 4: Parameters and statistics

Spread

(20 -29)^2+ (30 -29)^2 + (35 -29)^2 + (24 -29)^2 + (36 -29)^2 = 57,104

192 / (5-1) = 48 mm^2 Problem: weird units!

# A tibble: 1 × 2
   mean variance
  <dbl>    <dbl>
1    29       48

Lecture 4: Parameters and statistics

Spread

• Standard Deviation(σ, s): square root of variance.

  • In same units as observations

  • In example: √48 = 6.9 mm

• Coefficient of variation: SD as % of mean.
  • Useful for comparing spread in samples with different means
  • In example: (6.9/29)*100= 23.8 %

\(s^2 = \sqrt{\sum_{i=1}^{n} \frac{(y_i - \bar{y})^2}{n-1}}\)

\(\text{Coefficient of variation} = \frac{S}{\bar{Y}} \times 100\)

Lecture 4: Estimation

Problem: - don’t know the values of parameters

Goal: - estimate parameters from empirical data (samples)

3 general methods of parameter estimation: - Maximum Likelihood Estimation (MLE) - Ordinary Least Squares (OLS) - Resampling techniques

• MLE general method to estimate parameters in a way that maximizes the likelihood of the observed data given the parameter values.

• aims to find the parameter values that make the observed data most probable under the assumed statistical model.

• OLS specific method to estimate parameters of a linear regression model.

• minimizes the sum of the squared differences between observed and predicted values

Lecture 4: Estimation